On winning the penalty shoot-out in soccer.

The penalty shoot-out is used to break tied games in the knock-out stages of soccer competition. The shoot-out, which consists of an alternating series of penalty kicks, is won by the team with the highest goal tally after n kicks per team (n = 5). In the event of a tie after five penalty kicks each, the shoot-out progresses to 'sudden death' by increasing n in iterative fashion (i.e. n = n + 1) until one team obtains a higher goal tally than the other after an equal number of kicks per team. The team to strike first is determined at the end of extra time by the toss of a coin. As each on-field player can be awarded only a single penalty kick, the line-up order in which the penalty kicks are taken allows for the possibility of tactical influence on the final outcome. Consequently, we report a probability analysis of the penalty shoot-out in soccer from which we identify the following pre- and post-game strategies. The best five ranked penalty takers from the on-field players should be assigned to the first five penalty kicks in their reverse order of ability. That is, the fifth best penalty taker should take the first penalty kick, the fourth best penalty taker should take the second penalty kick, and so on. In the event of sudden death, the next highest ranked on-field player should be assigned to the next penalty kick until the shoot-out ends. For this tactic to be successful, players should be ranked a priori on their penalty-taking ability. Similarly, goalkeepers should be ranked a priori on their penalty-stopping ability. These findings indicate that the tactical substitution of on-field players for higher ranked penalty takers, including higher ranked penalty stoppers (i.e. goalkeepers), with a view to an impending penalty shoot-out should be given due consideration. These results are of practical importance in that they are shown to maximize the likelihood of winning the penalty shoot-out under certain initial conditions.